Is ${541686}$ divisible by $9$ ?
A number is divisible by $9$ if the sum of its digits is divisible by $9$ . [ Why? First, we can break the number up by place value: $ \begin{eqnarray} {541686}= &&{5}\cdot100000+ \\&&{4}\cdot10000+ \\&&{1}\cdot1000+ \\&&{6}\cdot100+ \\&&{8}\cdot10+ \\&&{6}\cdot1 \end{eqnarray} $ Next, we can rewrite each of the place values as $1$ plus a bunch of $9$ s: $ \begin{eqnarray} {541686}= &&{5}(99999+1)+ \\&&{4}(9999+1)+ \\&&{1}(999+1)+ \\&&{6}(99+1)+ \\&&{8}(9+1)+ \\&&{6} \end{eqnarray} $ Now if we distribute and rearrange, we get this: $ \begin{eqnarray} {541686}= &&\gray{5\cdot99999}+ \\&&\gray{4\cdot9999}+ \\&&\gray{1\cdot999}+ \\&&\gray{6\cdot99}+ \\&&\gray{8\cdot9}+ \\&& {5}+{4}+{1}+{6}+{8}+{6} \end{eqnarray} $ Any number consisting only of $9$ s is a multiple of $9$ , so the first five terms must all be multiples of $9$ That means that to figure out whether the original number is divisible by $9 $ , all we need to do is add up the digits and see if the sum is divisible by $9$ . In other words, ${541686}$ is divisible by $9$ if ${ 5}+{4}+{1}+{6}+{8}+{6}$ is divisible by $9$ Add the digits of ${541686}$ $ {5}+{4}+{1}+{6}+{8}+{6} = {30} $ If ${30}$ is divisible by $9$ , then ${541686}$ must also be divisible by $9$ ${30}$ is not divisible by $9$, therefore ${541686}$ must not be divisible by $9$.